The global operations are the matrix-vector products [??] and [??] in step 1, the

dot product [??], and an addition of vectors [??].

Weight and Hamming distance are treated in Section 3.2, the dot product in Section 3.3 and iteration over all elements of [F.sup.n.sub.3] in Section 3.4.

The elementwise multiplication VW should not be confused with the dot product V x W [??] [V.sup.(1)][W.sup.(1)] + ...

For the dot product, weight and the Hamming distance we also use 64-bit remainder, multiplication and shifts and a special 'population count' instruction, if available.

Furthermore, from inspection of the NRF, one may ascertain extra relationships (not required by one of the 44 reduced forms) among the

dot products. The experimentalist (or user of cell data in the crystallographic databases) should be aware that any extra specialization in the NRF may signify an important fact: for example, that one has inadvertently determined a derivative cell of a lattice of higher symmetry.

It is easy to verify that the vectors are unit length and that their dot products are exactly (-1/(k - 1)).

Given a graph G = (V, E) on n vertices, and a real number k [is greater than or equal to] 1, a vector k-coloring of G is an assignment of unit vectors [v.sub.i] from the space [R.sup.n] to each vertex i [element of] V, such that for any two adjacent vertices i and j the dot product of their vectors satisfies the inequality

For all positive integers k and n such that k [is less than or equal to] n + 1, there exist k unit vectors in [R.sup.n] such that the dot product of any distinct pair is -1/(k - 1).

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The function dot product DP has three steps operating on conceptual units, as well, and no step is repeated.

Moreover, its algorithm perfectly matches an interconnection mode configurable architecture since each dot product forms a reversed binary-tree graph.